Model a perpetuity, find the payment that keeps the balance constant, and relate it to the interest earned each period.

How a perpetuity pays a regular amount forever by withdrawing only the interest earned, how to find the sustainable payment or required principal, and how it differs from a draw-down annuity.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
What makes a perpetuity last forever
Finding the payment or the principal
Perpetuity versus ordinary annuity

What this dot point is asking

You must recognise a perpetuity, find the payment that keeps the balance constant, and find the principal needed to support a given payment forever.

What makes a perpetuity last forever

A perpetuity is a special draw-down annuity in which the regular withdrawal exactly equals the interest earned each period. Because only the interest is taken, the principal is untouched and the fund pays out indefinitely.

This is the steady-state of the draw-down recurrence $V_{n+1} = (1 + i)V_n - d$ : when $d = iV$ , the balance returns to $V$ each period.

Finding the payment or the principal

Two questions arise. Given a principal and rate, find the largest sustainable payment ( $d = iV$ ). Given a desired payment and rate, find the principal that must be invested ( $V = d/i$ ). Both come straight from the same relationship.

Worked example

Funding a scholarship forever

A donor wants to fund a \ $3000 annual scholarship forever from a fund earning$ 4%$ per year. How much must be invested?

Use $V = d/i = 3000 / 0.04 = 75\,000$ .

So \ $75\,000 must be invested. Check: interest in year one$ = 0.04 \times 75,000 = 3000 $, exactly the scholarship, so the \$ 75,000 principal is left intact and pays $3000 every year indefinitely.

If instead the donor invests \ $75\,000 but pays \$ 3500 a year, the extra $500 comes out of principal, the fund shrinks, and the scholarship cannot last forever; that is an ordinary draw-down annuity, not a perpetuity.

Perpetuity versus ordinary annuity

The dividing line is the interest. Withdraw exactly the interest and the fund is a perpetuity; withdraw more and it is a finite annuity that runs down. Recognising which case a question describes tells you whether to use $d = iV$ or to iterate the draw-down recurrence.